On the other hand, maybe what you want is a combinatorial interpretation. There is a way to express a related function as a binomial determinant which can be combinatorially interpreted using Gessel-Viennot, as described here.

Let Bk(z) denote the usual binomial coefficient, suitably generalized to allow z to be a formal variable. Assuming that you intend to have the same number of factors in the numerator and denominator in your definition of F(a,k)(z), then via the substitution z = aw,